The DIN-sync run signal isn't a pulse. When you press play the run pin will go high, and pressing stop will take it low again.
There a switch in the kit thats wired up to select between internal and external clock. When the switch is set to external sync and the DIN-sync run pin is high, the wires from the PCB to the play and stop buttons take the play button high and the stop button low. This starts playback in sync with the clock.
When the run pin goes low, the play button also goes low and the stop button goes high. This stops playback and resets the DR110 to the start of the bar.
If you set the switch to internal clock the play and stop buttons float, so they can then be pressed from the front panel without interference from the sync interface.
In external sync mode pressing stop while the DR110 is running will stop it, but pressing play won't do anything as it won't start playing again until the master clock restarts. If you leave the stop button wire disconnected the DR110 won't automatically reset when the master clock stops, so it'll just restart where it left off when the master restarts.